If the scalars αα are real,
SS is called a real vector
space.
If the scalars αα are complex,
SS is called a complex
vector space.
If the "vectors" in SS are functions
of a continuous variable, we sometimes call
SS a linear function
space
We define a set
V
V
to be a vector space if
-
x+y=y+x
x
y
y
x
for each
x
x and
y
y in
V
V
-
x+(y+z)=(x+y)+z
x
y
z
x
y
z
for each
x
x,
y
y, and
z
z in
V
V
-
There is a unique "zero vector" such that
x+0=x
x
0
x
for each
x
x in
V
V
(0 is the field additive identity)
- For each
x
x in
V
V there is a unique vector
−x
x
such that
x+−x=0
x
x
0
-
1x=x
1
x
x
(1 is the field multiplicative identity)
-
(
c
1
c
2
)
x=
c
1
(
c
2
x
)
(
c
1
c
2
)
x
c
1
(
c
2
x
)
for each
x
x in
V
V and
c
1
c
1
and
c
2
c
2
in
ℂ
ℂ
-
c(x+y)=cx+cy
c
x
y
c
x
c
y
for each
x
x and
y
y in
V
V and
c
c in
ℂ
ℂ
-
(
c
1
+
c
2
)x=
c
1
x+
c
2
x
c
1
c
2
x
c
1
x
c
2
x
for each
x
x in
V
V and
c
1
c
1
and
c
2
c
2
in
ℂ
ℂ
-
Rn=real vector space
n
real vector space
-
Cn=complex vector space
n
complex vector space
-
L
1
R=
ft
ft
∫−∞∞|ft|dt<∞
L
1
f
t
t
f
t
f
t
is a vector space
-
L
∞
R=
ft
ft
f
(
t
)
is bounded
L
∞
f
t
f
(
t
)
is bounded
f
t
is a vector space
-
L
2
R=
ft
ft
∫−∞∞|ft|2dt<∞
=finite energy signals
L
2
f
t
t
f
t
2
f
t
finite energy signals
is a vector space
-
L
2
0
T
=finite energy functions on interval [0,T]
L
2
0
T
finite energy functions on interval [0,T]
-
ℓ
1
Z
ℓ
1
,
ℓ
2
Z
ℓ
2
,
ℓ
∞
Z
ℓ
∞
are vector spaces
-
The collection of functions piecewise constant between the
integers is a vector space
-
ℝ
+
2=
x
0
x
1
x
0
x
1
(
x
0
>0) and (
x
1
>0)
ℝ
+
2
x
0
x
1
x
0
0
x
1
0
x
0
x
1
is not a vector space.
11∈
ℝ
+
2
1
1
ℝ
+
2
, but
α11∉
ℝ
+
2 ,
α<0
α
α
0
α
1
1
ℝ
+
2
-
D=z∈C ,
|z|≤1
D
z
z
1
z
is not a vector space.
(
z
1
=1)∈D
z
1
1
D
,
(
z
2
=j)∈D
z
2
D
, but
(
z
1
+
z
2
)∉D
z
1
z
2
D
,
|
z
1
+
z
2
|=2>1
z
1
z
2
2
1
Vector spaces can be collections of functions, collections
of sequences, as well as collections of traditional
vectors (i.e. finite lists of numbers)
"My introduction to signal processing course at Rice University."